F20MieScattering

Particle size determination using Mie scattering theory

Yunao Zheng

Abstract

This experiment aims to measure the size of polystyrene microspheres using Mie scattering theory. We measured the scattering intensity as a function of the scattering angle of a He-Ne laser scattered by the distilled water solution containing polystyrene particles with diameters of 3um and 5um, respectively. We fitted the measured angular scattering distribution to theoretical patterns generated using Prahl’s online calculator to determine the size of the particle. The best fit for 3um polystyrene was found to be 3.08um, and the best fit for 5um polystyrene was found to be 5.12um, the results are quite close to the nominal sizes from the manufacturer.

Introduction

Mie scattering describes the scattering when the wavelength of the incident light is close to the size of the scattering particle. It appears in many respects in our daily life. For example, it gives clouds a white appearance because the cloud droplets scatter all wavelengths of visible light containing in the sunlight^[1]. One application of the Mie scattering is using the laser-scattering technique to determine the size of the microscopic spherical particles^[2]. This method can be applied in many different areas, such as the study of biological tissues^[3], the measurement of dust-concentration^[4], the study of the metallic surface’s properties^[5], and the investigation of the planetary atmosphere^[6]. Compared to other sizing techniques like using electron microscopy to measure the particle size directly, the laser-light scattering technique using Mie scattering theory is more economical because it doesn’t require expensive apparatus^[2]. In this experiment, we will measure the angular scattering distribution of a He-Ne laser that incident upon the solution of polystyrene microspheres, and we determine the size of the spheres by fitting the experimental data into the theoretical distribution.

Theory

Mie theory gives the solution to Maxwell’s equation which describes the scattering of electromagnetic plane wave incident upon a single sphere. The solution was given by Weiner, Rust, and Donnelly^[2]. The electrical and magnetic fields outside the sphere are:

where N and M are the vector spherical harmonics that can be expressed by basis functions, and the coefficients are

where jn is the spherical Bessel function of the first kind, hn is the spherical Hankel function, μ and μ1 are the permeability of the surrounding and sphere medium, x is the size parameter. The size parameter is given by x =πNd/λ, where N is the refractive index of the surrounding medium, d is the diameter of the scattering sphere and λ is the wavelength of the incident light. Since all other terms are either constants or basis functions, the angular scatting distribution is dependent upon the size parameter x. If we fix N and λ, the distribution is dependent upon the diameter of the sphere. We can make some sample plots of the theoretical scattering intensities as a function of the scattering angle for different diameters.

Figure 1. The theoretical angular scattering distribution for particles of diameters of 0.75um, 3um, 5um. Data were calculated using Prahl's online calculator, under the condition N=1.33 and lambda=632.8nm.

We can see different particle sizes produce distinct scattering patterns. Therefore, we can determine the particle size by measuring its angular scattering distribution and fit the experimental data to the theoretical model.

Experimental Setup

Figure 2. A scheme of the experimental setup.

The He-Ne laser is linearly polarized, it has a power of 3mW and a wavelength of 632.8nm. The light path is aligned by mirrors to make the laser incidents upon the glass square cuvette fixed above the stepper motor’s axis. The cuvette contains the water solution of polystyrene microspheres, the water is distilled so no other particles can scatter the light. The laser will be scattered by spheres and generate a scattering pattern of concentric, alternating bright and dark rings. It is important to make sure the solution has a proper concentration, if the concentration is too high or too low, the scattering pattern will be hard to observe.

A half-inch photodetector is attached to the end of the arm of the stepper motor, this photodetector is connected to a digital voltmeter to measure light intensity the photodetector detected. The arm of the stepper motor can rotate in a plane about the stepper motor’s axis. By rotating the arm of the stepper motor, the photodetector can swipe through the scattering pattern in a plane and measure the light intensities in different areas of the pattern. The stepper motor has a step size of 0.45 degrees, so each time we rotate the arm for 0.45 degrees and measure the corresponding light intensity on the pattern. With the stepper motor, the photodetector is able to measure the light intensities on the pattern as a function of the scattering angle. The stepper motor and the digital voltmeter are connected to a computer, they are controlled by a LabView program, this program can record the measured data and write the data to a CSV file.

Result

We measured the scattering intensity as a function of scattering angle with 3um and 5um diameter polystyrene particles. Since a part of the light will pass through the cuvette without being scattered, it will cause a very large intensity at the middle of the pattern. It doesn’t agree with the theoretical Mie scattering pattern, so all data points that lower than 9 degrees were dropped. Also, we dropped all the data points greater than 36 degrees, since the scattering intensities after 36 degrees are very low that our photodetector is not able to distinguish them from the background noise. The distribution is symmetrical about angle=0, so we only use data on the right side to perform the fitting. We found the best fitting for the 3um and 5um polystyrenes are 3.08um and 5.12um, respectively.

Figure 3. The measured scattering data for polystyrene particle with a diameter of 3.0 micrometers in water solution. The best fit was found to be d= 3.08 micrometers. It has a reduced chi-square of 26.4827.

Figure 4. The calculated chi-square plot as a function of the diameter of the particle. We can find the minimum chi-square appears at d=3.08 um.

Figure 5. The measured scattering data for polystyrene particle with a diameter of 5.0 micrometers in water solution. The best fit was found to be d= 5.12 micrometers. It has a reduced chi-square of 70.8262.

Figure 6. The calculated chi-square plot as a function of the diameter of the particle. The minimum chi-square appears at d=5.12 um.

Conclusion

We found this method can give a result that pretty close to the nominal size, but the fittings are not very good due to the large reduced-chi-square. There are some possible factors that may responsible for the error in the result and the large reduced-chi-square. The scattered light is refracted when exits the cuvette and it affects the scattering distribution dramatically, but we only used an approximation method to correct it. The other thing that might responsible for the large-reduced chi-square is the convolution, because the photodetector has a width (half-inch), it is not measuring the intensity at a specific angle, but the average intensity within a range. If we can perform a deconvolution, if we can perform a deconvolve, we might get a result with lower reduced-chi-square. Also, the polarization of light and the concentration of the solution may matter, but it's hard to tell how they affect the result, both of them need further investigation.

Reference

[1] Mie Scattering. (n.d.). WW2010. Retrieved September 30, 2020, from http://ww2010.atmos.uiuc.edu/(Gl)/guides/mtr/opt/mch/sct/mie.rxml

[2] Weiner, I., Rust, M., & Donnelly, T. D. (2001). Particle size determination: An undergraduate lab in Mie scattering. American Journal of Physics, 69(2), 129–136. https://doi.org/10.1119/1.1311785

[3] Chorfi, H., Ayadi, K., & Boufendi, L. (2018). Light Scattering Applied to the Study of Biological Tissues: To an Optical Biopsy. 637–643. https://doi.org/10.1007/978-3-319-89707-3_67

[4] Yu, X., Shi, Y., Wang, T., & Sun, X. (2017). Dust-concentration measurement based on Mie scattering of a laser beam. PLoS ONE, 12(8), e0181575. https://doi.org/10.1371/journal.pone.0181575

[5] Monfared, S. K., Buttler, W. T., Frayer, D. K., Grover, M., LaLone, B. M., Stevens, G. D., Stone, J. B., Turley, W. D., & Schauer, M. M. (2015). Ejected particle size measurement using Mie scattering in high explosive driven shockwave experiments. Journal of Applied Physics, 117(22), 223105. https://doi.org/10.1063/1.4922180

[6] Benneke, B., Knutson, H. A., Lothringer, J., Crossfield, I. J. M., Moses, J. I., Morley, C., Kreidberg, L., Fulton, B. J., Dragomir, D., & Howard, A. W. (2019). A sub-Neptune exoplanet with a low-metallicity methane-depleted atmosphere and Mie-scattering clouds. Nat Astron, 3(9), 813–821. https://doi.org/10.1038/s41550-019-0800-5

[7] Prahl, S. (n.d.). Mie Scattering Calculator. OMLC. Retrieved October 1, 2020, from https://omlc.org/calc/mie_calc.html